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Monday, December 2, 2013

Many colleges and universities have a
mathematics or quantitative reasoning requirement that ensures that
no student graduates without completing at least one sufficiently
mathematical course.

Recognizing that taking a regular
first-year mathematics course—designed for students majoring in
mathematics, science, or engineering—to satisfy a QR requirement
is not educationally optimal (and sometimes a distraction for the
instructor and the TAs who have to deal with students who are neither
motivated nor well prepared for the full rigors and pace of a
mathematics course), many institutions offer special QR courses.

I’ve always enjoyed giving such
courses, since they offer the freedom to cover a wide swathe of
mathematics—often new or topical parts of mathematics. Admittedly
they do so at a much more shallow depth than in other courses, but a
depth that was always a challenge for most students who signed up.

Having been one of the pioneers of
so-called “transition courses” for incoming mathematics majors
back in the 1970s, and having given such courses many times in the
intervening years, I never doubted that a lot of the
material was well suited to the student in search of meeting a QR
requirement. The problem with classifying a transition course as a QR
option is that the goal of preparing an incoming student for the
rigors of college algebra and real analysis is at odds with the
intent of a QR requirement. So I never did that.

Enter MOOCs. A lot of the stuff that is
written about these relatively new entrants to the higher education
landscape is unsubstantiated hype and breathless (if not fearful)
speculation. The plain fact is that right now no one really knows
what MOOCs will end up looking like, what part or parts of the
population they will eventually serve, or exactly how and where they
will fit in with the rest of higher education. Like most others I
know who are experimenting with this new medium, I am treating it
very much as just that: an experiment.

The first version of my MOOC
Introduction to Mathematical Thinking, offered in the fall
of 2012, was essentially the first three-quarters of my regular
transition course, modified to make initial entry much easier,
delivered as a MOOC. Since then, as I have experimented with
different aspects of online education, I have been slowly modifying
it to function as a QR-course, since improved quantitative reasoning
is surely a natural (and laudable) goal for online courses with
global reach—that “free education for the world” goal is
still the main MOOC-motivator for me.

I am certainly not viewing my MOOC as
an online course to satisfy a college QR requirement. That may
happen, but, as I noted above, no one has any real idea what role(s)
MOOCs will end up fulfilling. Remember, in just twelve months,
the Stanford MOOC startup Udacity, which initiated all the media
hype, went from “teach the entire world for free” to
“offer corporate training for a fee.” (For my (upbeat) commentary
on this rapid progression, see my article in the Huffington Post.)

Rather, I am taking advantage of the
fact that free, no-credential MOOCs currently provide a superb
vehicle to experiment with ideas both for classroom teaching and for
online education. Those of us at the teaching end not only learn what
the medium can offer, we also discover ways to improve our classroom
teaching; while those who register as students get a totally free
learning opportunity. (Roughly three-quarters of them already have a
college degree, but MOOC enrollees also include thousands of
first-time higher education students from parts of the world that
offer limited or no higher education opportunities.)

The biggest challenge facing anyone who
wants to offer a MOOC in higher mathematics is how to handle the fact
that many of the students will never receive expert feedback on their
work. This is particularly acute when it comes to learning how to
prove things. That’s already a difficult challenge in a regular
class, as made clear in this great blog post by “mathbabe” Cathy O’Neil.
In a MOOC, my current view is it would be unethical to try. The last
thing the world needs are (more) people who think they know
what a proof is, but have never put that knowledge to the test.

But when you think about it, the idea
behind QR is not that people become mathematicians who can prove
things, rather that they have a base level of quantitative literacy
that is necessary to live a fulfilled, rewarding life and be a
productive member of society. Being able to prove something
mathematically is a specialist skill. The important general
ability in today’s world is to have a good understanding of the
nature of the various kinds of arguments, the special nature of
mathematical argument and its role among them, and an ability to
judge the soundness and limitations of any particular argument.

In the case of mathematical argument,
acquiring that “consumer’s understanding” surely
involves having some experience in trying to construct very simple
mathematical arguments, but far more what is required is being able
to evaluate mathematical arguments.

And that can be handled in a MOOC. Just
present students with various mathematical arguments, some correct,
others not, and machine-check if, and how well, they can determine
their validity.

Well, that leading modifier “just”
in that last sentence was perhaps too cavalier. There clearly is more
to it than that. As always, the devil is in the details. But once you
make the shift from viewing the course (or the proofs part of the
course) as being about constructing proofs to being about
understanding and evaluating proofs, then what
previously seemed hopeless suddenly becomes rife with possibilities.

I started to make this shift with the
last session of my MOOC this fall, and though there were significant
teething troubles, I saw enough to be encouraged to try it again—with modifications—to an even greater extent next year.

Of course, many QR courses focus on
appreciation of mathematics, spiced up with enough “doing math”
content to make the course defensibly eligible for QR fulfillment.
What I think is far less common—and certainly new to me—is
using the evaluation of proofs as a major learning vehicle.

What makes this possible is that the
Coursera platform on which my MOOC runs has developed a peer review
module to support peer grading of student papers and exams.

The first times I offered my MOOC, I
used peer evaluation to grade a Final Exam. Though the process worked
tolerably well for grading student mathematics exams—a lot better
than I initially feared—to my eyes it still fell well short of
providing the meaningful grade and expert feedback a professional
mathematician would give. On the other hand, the benefit to the
students that came from seeing, and trying to evaluate, the proof
attempts of other students, and to provide feedback, was significant—both in terms of their gaining much deeper insight into the
concepts and issues involved, and in bolstering their confidence.

When the course runs again in a few
week's time, the Final Exam will be gone, replaced by a new course
culmination activity I am calling Test Flight.

How will it go? I have no idea. That’s
what makes it so interesting. Based on my previous experiments, I
think the main challenges will be largely those of implementation. In
particular, years of educational high-stakes testing robs many
students of the one ingredient essential to real learning: being
willing to take risks and to fail. As young children we have it.
Schools typically drive it out of us. Those of us lucky enough to end
up at graduate school reacquire it—we have to.

I believe MOOCs, which offer community
interaction through the semi-anonymity of the Internet, offer real
potential to provide others with a similar opportunity to re-learn
the power of failure. Test Flight will show if this belief is
sufficiently grounded, or a hopelessly idealistic dream! (Test
flights do sometimes crash and burn.)

The more people learn to view failure
as an essential constituent of good learning, the better life will
become for all. As a world society, we need to relearn that innate
childhood willingness to try and to fail. A society that does not
celebrate the many individual and local failures that are an
inevitable consequence of trying something new, is one destined to
fail globally in the long term.

For those interested, I’ll be
describing Test Flight, and reporting on my progress (including the
inevitable failures), in my blog MOOCtalk.org
as the experiment continues. (The next session starts on February 3.)

Monday, November 4, 2013

The trouble with writing about, or quoting, Liping Ma, is
that everyone interprets her words through their own frame, influenced by their
own experiences and beliefs.

“Well, yes, but isn’t that true for anyone reading
anything?” you may ask. True enough. But in Ma’s case, readers often arrive at
diametrically opposed readings. Both sides in the US Math Wars quote from her
in support of their positions.

Still, if I stopped and worried about
readers completely misreading or misinterpreting things I write, Devlin’s Angle would likely appear maybe
once or twice a year at most. So you can be sure I am about to press ahead and
refer to her recent article regardless.

My reason for doing so is that I am largely
in agreement with what I believe she is saying. Her thesis (i.e., what I
understand her thesis to be) is what lay behind the design of my MOOC and my recently released video game. (More on both later.)

Broadly speaking, I think most of the furor
about K-12 mathematics curricula that seems to bedevil every western country
except Finland is totally
misplaced. It is misplaced for the simple, radical (except in Finland) reason that curriculum doesn’t really matter. What matter are teachers. (That last sentence
is, by the way, the much sought after “Finnish secret” to good
education.) To put it simply:

BAD CURRICULUM + GOOD OR WELL-TRAINED
TEACHERS = GOOD EDUCATION

GOOD CURRICULUM + POOR OR POORLY-TRAINED
TEACHERS = POOR EDUCATION

I am very familiar
with the Finnish education system. The Stanford H-STAR institute I co-founded and direct has been collaborating with Finnish
education researchers for over a decade, we host education scholars from
Finland regularly, I travel to Finland several times a year to work with
colleagues there, I am on the Advisory
Board of CICERO
Learning, one of their leading educational research
organizations, I’ve spoken with members of the Finnish government whose focus
is education, and I’ve sat in on classes in Finnish schools. So I know from
firsthand experience in the western country that has got it right that teachers
are everything and curriculum is at most (if you let it be) a distracting
side-issue.

The only people for
whom curriculum really matters are politicians and the politically motivated
(who can make political capital out of curriculum) and publishers (who make a
lot of financial capital out of it).

But I digress:
Finland merely serves to provide an existence proof that providing good
mathematics education in a free, open, western society is possible and has
nothing to do with curriculum. Let’s get back to Liping Ma’s recent Notices article. For she provides a
recipe for how to do it right in the curriculum-obsessed, teacher-denigrating US.

Behind Ma’s
suggestion, as well as behind my MOOC and my video game (both of which I have
invested a lot of effort and resources into) is the simple (but so often
overlooked) observation that, at its heart, mathematics is not a body of facts
or procedures but a way of thinking.
Once a person has learned to think that way, it becomes possible to learn and
use pretty well any mathematics you need or want to know about, when you need
or want it.

In principle, many areas of mathematics can be used to master that way of thinking, but
some areas are better suited to the task, since their learning curve is much
more forgiving to the human brain.

For my MOOC, which
is aimed at beginning mathematics students at college or university, or high
school students about to become such, I take formalizing the use of language
and the basic rules of logical reasoning (in everyday life) as the subject matter, but the focus is as described in the last two
words of the course’s title: Introduction
to Mathematical Thinking.

Apart from the final
two weeks of the course, where we look at elementary number theory and
beginning real analysis, there is really no mathematics in my course in the
usual sense of the word. We use everyday reasoning and communication as the
vehicle to develop mathematical thinking.

[SAMPLE PROBLEM:
Take the famous (alleged) Abraham Lincoln quote, “You can fool all of the
people some of the time and some of the people all of the time, but you cannot
fool all the people all the time.” What is the simplest and clearest positive
expression you can find that states the negation of that statement? Of course,
you first have to decide what “clearest”, “simplest”, and “positive” mean.]

Ma’s focus in her
article is beginning school mathematics. She contrasts the approach used in
China until 2001 with that of the USA. The former concentrated on “school
arithmetic” whereas, since the 1960s, the US has adopted various instantiations
of a “strands” approach. (As Ma points out, since 2001, China has been moving
towards a strands approach. By my read of her words, she thinks that is not a
wise move.)

In principle, I find
it hard to argue against any of these—provided
they are viewed as different facets of a single whole.

The trouble is, as
soon as you provide a list, it is almost inevitable that the first system
administrator whose desk it lands on will turn it into a tick-the-boxes
spreadsheet, and in turn the textbook publishers will then produce massive (hence
expensive) textbooks with (at least) ten chapters, one for each column of the
spreadsheet. The result is the justifiably maligned “Mile wide, inch deep” US
elementary school curriculum.

It’s not that the
idea is wrong in principle. The problem lies in the implementation. It’s a long
path from a highly knowledgeable group of educators drawing up a curriculum to
what finds its way into the classroom—often to be implemented by teachers
woefully unprepared (through no fault of their own) for the task, answerable to
administrators who serve political leaders, and forced to use textbooks that
reinforce the separation into strands rather than present them as variations on
a single whole.

Ma’s suggestion is
to go back to using arithmetic as the primary focus, as was the case in Western
Europe and the United States in the years of yore and China until the turn of the
Millennium, and use that to develop all of the mathematical thinking skills the
child will require, both for later study and for life in the twenty-first
century. I think she has a point. A good point.

She is certainly not
talking about drill-based mastery of the classical Hindu-Arabic algorithms for
adding, subtracting, multiplying, and dividing, nor is she suggesting that the
goal should be for small human beings to spend hours forcing their analogically
powerful, pattern-recognizing brains to become poor imitations of a ten-dollar
calculator. What was important about arithmetic in past eras is not necessarily
relevant today. Arithmetic can be used to trade chickens or build spacecraft.

No, if you read what
she says, and you absolutely should,
she is talking about the rich, powerful structure of the two basic number
systems, the whole numbers and the rational numbers.

Will that study of
elementary arithmetic involve lots of practice for the students? Of course it
will. A child’s life is full of practice. We are adaptive creatures, not
cognitive sponges. But the goal—the motivation for and purpose of that
practice—is developing arithmetic
thinking, and moreover doing so in a manner that provides the foundation
for, and the beginning of, the more general mathematical
thinking so important in today’s world, and hence so empowering for today’s
citizens.

The whole numbers
and the rational numbers are perfectly adequate for achieving that goal. You
will find pretty well every core feature of mathematics in those two systems.
Moreover, they provide an entry point that everyone is familiar with, teacher,
administrator, and beginning elementary school student alike.

In particular, a
well trained teacher can build the necessary thinking skills and the
mathematical sophistication —and cover whatever strands are in current favor—without having to bring in any other mathematical structure.

When you adopt the
strands approach (pick your favorite flavor), it’s very easy to skip over
school arithmetic as a low-level skill set to be “covered” as quickly as
possible in order to move on to the
“real stuff” of mathematics. But Ma is absolutely right in arguing that this is
to overlook the rich potential still offered today by what are arguably (I would
so argue) the most important mathematical structures ever developed: the whole
and the rational numbers and their associated elementary arithmetics.

For what is often
not realized is that there is absolutely nothing elementary about elementary
arithmetic.

Incidentally, for my
video game, Wuzzit Trouble, I took whole number arithmetic and built a game around it. If
you play it through, finding optimal solutions to all 75 puzzles, you will find
that you have to make use of increasingly sophisticated arithmetical reasoning.
(Integer partitions, Diophantine equations, algorithmic thinking, and
optimization.)

I doubt Ma had video
game instantiations of her proposal in mind, but when I first read her article,
almost exactly when my game was released in the App Store (the Android
version came a few weeks later) that’s exactly what I
saw.

Other games my colleagues and I have designed but not yet built are based on different parts of mathematics. We started with one built around elementary arithmetic because arithmetic provides all the richness you need to develop mathematical thinking, and we wanted our first game to demonstrate the potential of game-based learning in thinking-focused mathematical education (as opposed to the more common basic-skills focus of most mathematics-educational games). In starting with an arithmetic-based game, we were (at the time unknowingly) endorsing the very point Ma was to make in her article.

Tuesday, October 1, 2013

In last
month’s column, I reflected on how modern technology enables one person—in my case an academic—to launch enterprises with (potential) global reach
without (i) money and (ii) giving up his day job. That is true, but
technology does not replace expertise and its feeder, experience.

In the case of my MOOC, now well into its
third offering, I’ve been teaching transition courses on mathematical thinking
since the late 1970s, and am able to draw on a lot of experience as to the
difficulties most students have with what for most of them is a completely new
side to mathematics.

Right now, as we get into elementary, discrete number theory, the class (the 9,000 of 53,000 registrants still active) is struggling to
distinguish between division—a binary operation on rationals that yields a
rational number for a given pair of integers or rationals—and divisibility—a relation between pairs
of integers that is either true or false for any given pair of integers. Unused
to distinguishing between different number systems, they suddenly find
themselves lost with what they felt they knew well, namely elementary arithmetic.

Anyone who has taught a transition course will be familiar
with this problematic rite of passage. I suspect I am not alone in having
vivid memories of when I myself went through it, even though it was many
decades ago!

As a result of all those years teaching this kind of
material, I pretty well know what to expect in terms of student difficulties
and responses, so can focus my attention on figuring out how to make it work in
a MOOC. I know how to filter and interpret the comments on the discussion
forum, having watched up close many generations of students go through it. As a
result, doing it in a MOOC format with a class spread across the globe is a
fascinating experiment, when it could so easily have been a disaster.

My one fear is that, because the course pedagogy is based on
Inquiry-Based Learning,
it may be more successful with experienced professionals (of whom I have many
in the class), rather than the course’s original target audience of recent high
school graduates. In particular, I suspect it is the latter who constantly
request that I show them how to solve a problem before expecting them to do so.
If all students have been exposed to is instructional teaching, and they have never
experienced having to solve a novel problem—to figure it out for themselves—it is probably unrealistic to expect them to make that leap in a Web-based
course. But maybe it can be made to work. Time will tell.

The other startup I wrote about was my video game company.
That is a very different experience, since almost everything about this is new
to me. Sure, I’ve been studying and writing about video game learning for many
years, and have been playing video games for the same length of time. But designing
and producing a video game, and founding a company to do it, are all new. Although
we describe InnerTube Games as “Dr. Keith Devlin’s video game company,” and most of the reviews of our first release
referred to Wuzzit Trouble as “Keith Devlin’s mathematics video game,” that was
like referring to The Rolling Stones as “Mick Jagger’s rock group.” Sure he was
out in front, but it was the entire band that gave us all those great
performances.

In reality, I brought just three new things to our video
game design. The first is our strong focus on mathematical thinking (the topic
of my MOOC) rather than the mastery of symbolic skills (which is what 99% of
current math ed video games provide). The second is that the game should embed
at least one piece of deep, conceptual mathematics. (Not because I wanted the
players to learn that particular piece of mathematics. Rather that its presence
would ensure a genuine mathematical
experience.) The third is the design principle that the video game should be
thought of as an instrument on which you “play math,” analogous to the piano, an instrument on which you play music.

In fact, I was not alone among the company co-founders in favoring
the mathematical thinking approach. One of us, Pamela, is a former
middle-school mathematics teacher and an award winning producer of educational
television shows, and she too was not interested in producing the 500th
animated-flash-card, skills-mastery app. (Nothing wrong
with that approach, by the way. It’s just that the skills-mastery sector is
already well served, and we wanted to go instead for something that is woefully
under-served.) I may know a fair amount about mathematics and education, and I
use technology, but that does not mean I'm an expert in the use of various media
in education. But Pamela is.

And this is what this month’s column is really about: the
need for an experienced and talented team to undertake anything as challenging
as designing and creating a good educational learning app. Though I use my own
case as an example, the message I want to get across is that if, like me, you
think it is worthwhile adding learning apps and video games to the arsenal of
media that can be used to provide good mathematics learning, then you need to
realize that one smart person with a good idea is not going to be anything like
enough. We need to work in teams with people who bring different expertise.

I’ve written extensively in my blog profkeithdevlin.org about the problems
that must be overcome to build good learning apps. In fact, because of the
history behind my company, we set our bar even higher. We decided to create
video games that had all the features of good commercial games developed for
entertainment. Games like Angry Birds
or Cut the Rope, to name two of my
favorites. Okay, we knew that, with a mathematics-based game, we are unlikely
to achieve the dizzying download figures of those industry-leading titles. But
they provided excellent exemplars in game structure, game mechanics, graphics,
sounds, game characters, etc. In the end, it all comes down to engagement,
whether the goal is entertainment and making money or providing good learning.

In other words, we saw (and see) ourselves not as an
“educational video game company” but as a “video game company.” But one that
creates video games built around
important mathematical concepts. (In the case of Wuzzit Trouble, those concepts are integer arithmetic, integer
partitions, and Diophantine equations.)

Going after that goal requires many different talents. I’ve
already mentioned Pamela, our Chief Learning Officer. I met her, together with
my other two co-founders, when I worked with them for several years on an
educational video game project at a large commercial studio. That project never
led to a released product, but it provided all four of us with the opportunity
to learn a great deal about the various crucial components of good video game
design that embeds good learning. Enough to realize, first, that we all needed
one another, and second that we could work well together. (Don’t underestimate
that last condition.)

By working alongside video game legend John Romero, I
learned a lot about what it takes to create a game that players will want to
play. Not enough to do so myself. But enough to be able to work with a good
game developer to inject good mathematics into such a game. That’s Anthony, the
guy on our team who takes a mathematical concept and turns it into a compelling
game activity. (The guy who can give me three good reasons why my “really cool
idea” really won’t work in a game!) Pamela, Anthony, and I work closely
together to produce fun game activities that embed solid mathematical learning,
each bringing different perspectives. Take any one of us out of the picture, and
the resulting game would not come close to getting those great release reviews
we did.

And without Randy, there would not even be a game to get
reviewed! Video games are, after all, a business. (At some point, we will have
to bring in revenue to continue!) The only way to create and distribute
quality games is to create a company. And yes, that company has to create and
market a product—something that’s notoriously difficult. (Google “why video
game companies fail.”) Randy (also a former teacher) was the overall production
manager of the project we all worked on together, having already spent many
years in the educational technology world. He’s the one who keeps everything
moving.

Like it or not, the world around us is changing rapidly, and
with so many things pulling on our students’ time, it’s no longer adequate to
sit back on our institutional reputations and expect students to come to us and
switch off the other things in their lives while they take our courses.

One case: I cannot see MOOCs replacing physical classes with
real professors, but they sure are already changing the balance. And you don’t
have to spend long in a MOOC to see the similarities with MMOs (massively
multiplayer online games).

We math professoriate long ago recognized we needed to
acquire the skills to prepare documents using word processing packages and
LaTeX, and to prepare Keynote or PowerPoint slides. Now we are having to learn
the rudiments of learning management systems (LMSs), video editing, the
creation of applets, and the use of online learning platforms.

Creating video games is perhaps more unusual, since it requires
so many different kinds of expertise, and I am only doing that because a
particular professional history brought me into contact with the gaming
industry. But plenty of mathematical types have created engaging math learning
apps, and some of them are really very good.

Technology not only makes all of these developments
possible, it makes it imperative that, as a community, we get involved. But in
the end, it’s people, not the technology, that make it happen. And to be
successful, those people may have to work in collaborative teams.

Wednesday, September 4, 2013

Last week turned out to be far more hectic than most, with
the simultaneous launch of two startups I have been involved in for the past few
years.

When I went into the life of academic mathematics some 42
years ago, I could never have imagined ever writing such a sentence. Nor, for
that matter, would I have had the faintest idea what a “startup” was. It’s a
measure of how much society has changed since 1971, when I transitioned from
being a “graduate student” to a “postdoc,” that today everyone knows what a
startup is, and many of my doctorate-bearing academic colleagues have, as a
sideline to their academic work, started up labs, centers, or companies. What
was once exceptional is now commonplace.

Massive changes in technology have made it, while not
exactly easy, at least possible for
anyone in academia to become an “edupreneur,” to use (just once, I promise) one
of the more egregious recent manufactured words. This means that, when our
academic work leads to a good idea or a product we think could be useful to
many of our fellow humans, we don’t have to sit back and hope that one day
someone will come along and turn it into something people can access or use. We
can make it available to them ourselves.

MOOCs are one of the most recent examples. If any of us in
the teaching business finds we have developed a course that students seemed to
have benefited from and we are proud of, we can (at least to some extent)
bottle it and make it available to a much wider audience.

Of course, we have had versions of that ability since the invention
of the printing press. Today, millions of people, academics and non-academics
alike, use those printing press descendants, websites and blogs, to achieve a
much wider audience for their written word.

A somewhat smaller (but growing) number have used platforms
such as YouTube and Vimeo to make video-recordings of their lectures widely
available.

To some extent, MOOCs can be viewed as an extension of both
of those Internet media developments. A MOOC sets out to achieve the very
ambitious goal of bottling an entire
college course and making it available to the entire world—or at least,
that part of the world with broadband access.

The launch this past weekend of the third iteration of my
constantly-evolving MOOC on Mathematical
Thinking was one of the two startups that gobbled up massive amounts of my
time over the past few weeks. Even though, having given essentially the same
course twice before, the bulk of the preparatory work was done, implementing
the changes I wanted to make and re-setting all the item release dates/times and
the various student submission deadlines was still a huge undertaking. For with
a MOOC, pretty well everything for the entire course needs to be safely deposited on (in my case, with my MOOC on Coursera) Amazon’s servers before the first of
my 41,000 registered students logged on over the weekend.

When you think about it, the very fact that a single
academic can do something like this, is pretty remarkable. What makes it
possible is that all the components are readily available. To go into the MOOC
business, all you need is a laptop, a word processor (and LaTeX, if you are
giving a math course), possibly a slide package such as PowerPoint, some kind
of video recording device (I use a standard, $900 consumer camcorder, others
use a digital writing tablet), a small microphone (possibly the one already
built in to your laptop), and a cheap consumer video editing package (I use
Premiere Elements, which comes in at around $90). Assuming you already have the
laptop and a standard office software package, you can set up in the MOOC
business for about $1500.

Sure, it helps if your college or university gives you
access to the open source MOOC platform edX, or is willing to enter an
agreement with, say, the MOOC platform provider Coursera. But if not, there are options such as YouTube, websites, Wikis, and blogs, all freely available.

My second startup was supposed to launch at least a month
before my MOOC, but a major hacking event at Apple’s Developer Site delayed their release of the first (free) mathematical thinking mobile game designed by
my small educational software company, InnerTube
Games. Both launches falling in the same week is not something I’d want to
do again!

Why form a company to create and distribute mathematics
education video games that incorporate some of the findings and insights I’d
developed over several years of research? The brutal answer is, I had no other viable
option. Though several years of research had convinced me that it was possible
to design and build “instruments” on which you can “play” parts of mathematics,
in the same way a musical instrument such as a piano can be used to play music
(in both cases by passing the need for static symbolic representations on a
page, which are known to be a huge barrier to learning for many people), I
simply was not successful in convincing funders it was a viable approach.

Clearly then, I had to build at least one such instrument. More
precisely, I had to team up with a small number of friends who brought the
necessary expertise I did not have. Again, a few years ago, it would have been
impossible for an academic to found and build a small company and create and
launch a product in my spare time. But today, anyone can.

Sure, even more so than with MOOCs, to form and operate an
educational software company, you need to work with other people—three in my
case. (That, at least, has been my experience.) But the key point is, the
technology and the resources infrastructure make it possible. You don’t have
to give up your day job as an academic to do it! And just as a MOOC provider
(or a YouTube, website, blogging platform combo) takes care of the distribution
of your course, so too the Web (in my case, in the form of Apple’s App Store)
can make your creation available to the world. At no cost.

We are not talking about enterprises designed with the
purpose of making money here—I am essentially in the same game as the
writing of academic works or textbooks, and in my case less so, since my books
cost money but my MOOC and my game are free. Rather we are making use of a
global infrastructure to make our work widely accessible. If that
infrastructure involves for-profit MOOC platforms or software companies, so be
it.

The fact is, it has never been as easy as it is today for
each one of us to take an idea or something we have created and make it
available to a wide audience. Sure, for both my examples, I have left a lot
unsaid, focusing on one particular aspect. (Take a glance at my video game
website to see who else was involved in that particular enterprise and the
experience they brought to the project. That was a team effort if ever there
was!) But the key fact is, it is now possible!

For more about my MOOC, and MOOCs in general, see my blog MOOCtalk.org. For my findings and thoughts on
mathematics education, see many of the posts on my other blog profkeithdevlin.org together with some
of the articles and videos linked to on the InnerTube
Games website.

And for another (dramatic) example of how one person with a
good idea can quickly reach a global audience, see Derek Muller’s superb STEM
education resource Veritasium.

Thursday, August 1, 2013

Readers who follow me on Twitter
will have noticed many tweets on the recent revelations about illegal
NSA surveillance. Here is why I think that none of us in mathematics
and mathematics education can ignore that debate.

There’s a popular conception that
mathematicians are unworldly, and that mathematics is, at its heart,
walled off from the real world, its pursuit a form of escapism that
takes the pursuer into a realm of pure, abstract thoughts.

Certainly, that’s a general sense of
mathematics that I held for many years. Yes, like all my fellow
mathematicians, I always knew that mathematics – all of it –
arose, directly or indirectly, from real world problems, and that any
branch of mathematics having any discipline-internal significance
almost always turns out to have real-world applications. But neither
of those was why I did mathematics. For most of my life as a
mathematician, I simply did not care about the history or application
of what I was doing. It was all about the chase – the search for
new knowledge in a beautiful domain.

Early on in my career, when more
politically active colleagues urged me to boycott conferences and
workshops funded by NATO (a big issue back in the 1970s), or to avoid
applying for research funds from commercial or military sources, I
essentially turned a deaf ear to what they were saying, and got on
with the work that interested me.

As a mathematician working in axiomatic
set theory, with particular foci on the properties of sets of large
infinite cardinality and on undecidability proofs, I felt fairly
confident that nothing I did would ever find practical application,
so for me the issue was purely one of where the money came from to
support my research. I felt “clean,” and not under any moral
pressure regarding potential unethical uses being made of my work.

True, I was aware that the famous early
twentieth century mathematician G. H. Hardy had made the same claim
about his work in number theory, yet in the mid-1970s his work found
highly significant application in the design of secure cryptographic
systems. But I felt that a similar outcome was unlikely in the case
of infinitary set theory. (I am no longer quite as sure about that; I
speculated about possible applications of Cantor’s set theory in my
June
column.)

I think we all have to address the
morality-of-possible-applications question about our work as
mathematicians at one time or another. Some, from Archimedes to Alan
Turing, have actively engaged in military research; others try to
avoid any direct contact with commercial or warfare-related
activities.

The rise of math-based corporations
such as Google that form a large and influential part of today’s
global world, and the closely related growth of the modern,
math-driven security state, as iconicized by the NSA, make it
impossible to maintain any longer the fiction (for such it always
was) that we can pursue mathematics as a pure activity, separate from
applications, be they good or ill.

The uncomfortable fact is, we are in no
different a situation than manufacturers of sporting guns who deny
any agency when their product is used to kill people. (Yes, people
pull the trigger, but as comedian Eddie Izzard pointed
out, “the gun helps.”)

If we want to be able to maintain that
our work will not play a role in someone’s death, torture, or
incarceration – or in someone else achieving enormous wealth and
power – our only option is to not go into mathematics in the first
place. The subject is simply way too powerful as a force – for good
or for evil.

Shortly after September 11, 2001, I was
asked to join a research project funded by the U.S. intelligence
service. For me, that was my crunch time. The work that led to that
invitation was an outgrowth (described in my 1995 book Logic
and Information) of my earlier research in
mathematical logic and set theory. Like it or not, I was already in
deep. To say no to that invitation would have been every bit a
positive action as to say yes. Sitting on the fence was not a
possibility. I was a mathematician. I’d already made the gun.

As the Google founders Larry Page and
Sergei Brin eventually discovered, “Do no evil” is a wonderful
grounding principle, but the power of mathematics renders it an
impossible goal to achieve. The best we can do is try to make our
voice heard, as many mathematicians and nuclear physicists did during
the Cold War, who spoke publicly about the massive scale of the
danger raised by nuclear weapons.

Finding out (as I have over the past
few weeks) that the work I’d done over the past twelve years –
for various branches of the U.S. government (intelligence and military)
and for commercial enterprises (in my case, the video game industry)
– was part of a body of research that had been subverted (as I see
it) to create a massive global surveillance framework, I felt I could
not remain silent.

Not because I felt that I, as an
individual, did anything of significance. I worked on non-classified
projects, and made no major breakthroughs. I was a very tiny cog in a
very big machine. (If “they” are keeping an eye on me, they are
definitely wasting our tax dollars!)

But I did take the money and I did do
the work. I don’t regret doing so. The fact is, I’d made the
crucial choice long before 2001; back in my youth when I decided to
become a mathematician.

Those of us in mathematics education
have always told our students that math is useful. In today’s world
more than ever, we cannot at the same time pretend it is free of
moral issues. Agnosticism is not an option (if it ever really was).
To say or do nothing is inescapably a positive act, just as
significant as saying or doing something.

We humans have created our mathematics,
and used it to help shape our world. Now we have to live in it. Not
only are we the ones who bear a large responsibility for that world,
we are also, by our very expertise, the ones who (in many fundamental
ways) understand it best. (It often seems that only the
mathematically sophisticated really appreciate that an American is
more likely to die in his or her bathtub than from a terrorist
attack, and that more people died on the roads due to increased
traffic during the time after 9/11 when all flights were grounded
than did in the Twin Towers attack.)

So, to return to the question implicit
in my title, “What is mathematics used for?” Douglas
Adams provided the answer: “Life, the
universe, everything.” With such reach and power comes responsibility.

FOOTNOTE: For a more personal take on
the above issues, see the interview
I did on June 21 on Shecky Riemann’s Math Tango blog.

Tuesday, July 2, 2013

When tech folk dabble in education (and tech writers cover it), the
excess of hype is sometimes matched only by their breathtaking lack of
knowledge about education. Even so, the above headline to the July 1 post by Forbes
contributor Jordan Shapiro must rank as one of the most stupid and ignorant
statements in human history.

It would be somewhat less ludicrous, though still open to debate, if
the headline had said “learn some algebra.” But “algebra”? All of
it?

Almost certainly, Shapiro himself did not write the headline—writers
rarely do. In fact, the article itself is fine. I have no problem with what
Shapiro wrote. But the fact that the ludicrous headline had not been changed 24
hours later indicates that Forbes’ editors feel happy with it. Sigh.

What the article itself reports is that, on average, students who
played a particular video game (DragonBox, of which more
later) completed a sufficient part of it in 42 minutes. Since the game itself
is based on algebraic principles, they could, therefore, be said to have
engaged in algebraic thinking. (I would be inclined to say just that, though
with any kind of machine learning—and human teaching if the instructor is not
paying close attention—one should always be on the lookout for an instance of
Benny’s Rules.)

Whether such performance in a video game justifies saying that the
students learned some (!) algebra in 42 minutes depends on what
metric you use to determine what learning has taken place.

Of course, if you define algebra to be (or to include) symbolic
manipulation, then successful completion of any video game is not going to
count as “doing algebra.” That is why I used the term “algebraic thinking” a
couple of paragraphs back. (See my previous blog post What is Algebra? for a
discussion of the distinction.) But is that the appropriate measure? What do we
want K-12 students to learn under the title “algebra”?

[ASIDE: There is another definitional question as to the classification
of DragonBox as a video game. Game developers have different views as to what
constitutes a video game. Some would describe DagonBox as an entertaining,
interactive, digital app, but would stop short at classifying it as a game.]

Before I go any further, I should give some disclaimers. First, as
readers of my blog profkeithdevlin.org (or my book Mathematics
Education for a New Era) will be aware, I am a strong proponent of the
use of video games in mathematics education. In fact, I advocate an approach to
the design of math ed video games that definitely includes DragonBox. I’ve met
the developer, Jean-Baptiste Huynh, and one of the co-founders of his company WeWantToKnow, and I used their game as an example in a
feature article on math ed
video games I wrote for American
Scientist in March of this year. I am about three-quarters of the way
through the second, greatly expanded version of the game, DragonBox2. Among the
designs for math ed video games that my own company, InnerTube Games, has been working on for several years,
are a couple that have much in common with DragonBox. (We are due to release
our first one, Wuzzit Trouble, this summer, but chose one based on arithmetic
and number theory to be our initial release, with algebra-based games to come
later.) So I am not a dispassionate outsider here.

For his Forbes article, Shapiro interviewed Jean-Baptiste Huynh, and
everything the DragonBox designer says, I agree with 100%. Here is my take on
the benefits of playing DragonBox (besides the fact that is it fun).

A student who plays through the new, greatly expanded version of the
game will undoubtedly engage in a substantial amount of (contextualized) algebraic
thinking focused on the solution of linear equations in one variable. The score
they obtain in the game will provide a good measure of how well they have
mastered that form of thinking (i.e., solving single-variable linear equations).

Does that mean the student can then sit down and ace a standard
written algebra exam? Not at all. Even though the later stages of DragonBox and
DragonBox2 involve on-screen manipulations of the standard symbolic
representations of equations, the step from physically moving digital objects
to manipulating symbolic expressions on a page is a much harder cognitive
challenge than one might first think. The human mind simply finds it very
difficult to reason in a purely abstract fashion. (In my book The Math Gene, published in
2000, I investigated the reasons for that difficulty.)

At issue is the notorious transfer problem, which, roughly
speaking, is the difficulty humans face in taking something that has been
learned in one context and applying it in another.

Huynh is of the opinion that it requires a human teacher to help the
student take the difficult step from completion of his game to mastery of
symbolic algebra, and I agree with him. I suspect that not everyone will be
able to make the transition, no matter how good the teaching, but many will.

There is certainly a lot to be gained from mastery of symbolic
algebra. First of all, learning at that level of abstraction is readily
applicable to any specific domain. Second, being able to reason free of the
complexities of any application domain is extremely powerful.

On the other hand algebra (or, more accurately, algebraic thinking)
was successfully used in commerce for many hundreds of years before the modern,
symbolic variety was introduced in the sixteenth century. So acquiring useful
algebra skills is not totally dependent on mastery of symbolic algebra.

A major question is, will playing DragonBox increase the likelihood
that a student will be able to master symbolic algebra, compared with a student
who does not have that game experience? There is good reason to assume the
answer is “Yes,” but that remains to be fully tested—something that can be
done only now the game (and others like it) is out. (The analogous question
remains to be answered for my own company’s forthcoming games.)

My reason for suspecting that playing video games like DragonBox is
highly beneficial in learning symbolic mathematics—the kind that is tested in
our school system—is perhaps best explained by an analogy from Hollywood. In
the 1984 movie The Karate Kid (I can’t bring
myself to watch the 2010 remake) and its sequel (KK2), martial arts instructor
Mr Miyagi prepares his
young pupil Daniel for Karate tournaments by getting him to polish a car, sand a
floor, catch a fly with chopsticks, and paint a fence, all of which develop the
reflexes and muscle memory required for key Karate moves, which Daniel uses to
great effect later in the movies.

True, this is not sound educational theory, though many teachers (and
most athletic coaches) adopt a similar approach. (This is a blog, remember, not
a research journal.) But until we have something more concrete, the analogy
works for me. Indeed, I am betting my company on it—as is Jean-Baptiste
Huynh.

Monday, June 3, 2013

We really have no way of knowing what
early humans thought when they gazed up at the sky. Since everyday
practical experience is, by definition, limited to a very small
region of space and time, it requires considerable cognitive
sophistication to conceive of something – say the night sky –
“going on for ever,” let alone to ponder whether that means it is
“infinite,” or indeed what “infinite” actually means.

What we do know is that the ancient
Greeks made what may have been the first substantial attempt to
analyze the notion of infinity, with Zeno of Elea (ca 490-430 BCE) of
particular note for his discussion of a number of (seeming) paradoxes
that arise from the assumption that space and time are (or are not)
infinitely divisible.

Archimedes’ (ca 287-112 BCE)
calculations of areas and volumes made implicit use
of infinity, and from today’s perspective can be recognized as the
forerunner of integral calculus.

Skillful formal –
though by modern standards not rigorous – use of the infinitely
large and the infinitely small was made by Isaac Newton and Gottfried
Leibniz in their development of modern infinitesimal calculus in the seventeenth century, though it was not until the nineteenth century
when Bernard
Bolzano, Augustin-Louis
Cauchy, and Karl
Weierstrass finessed the lurking problems of infinity by means of
the famous (and for many a first-year mathematics major, infamous)
epsilon-delta definitions of limits and continuity.

But none of these
developments was about infinity as an entity;
the focus rather was on the unending nature of certain processes,
starting with counting. It wasGeorg
Cantor (1845 – 1918) who really tackled infinity head on. His
proof that the set of real numbers cannot be put into one-one
correspondence with the natural numbers, and hence is of a larger
order of infinitude, led to a series of papers, published in a
remarkable ten-year period between 1874 and 1884, that formed the
basis for modern abstract set theory, including the development of a
fully formed arithmetical theory of infinite numbers (or
“cardinals”).

Reactions to Cantor’s revolutionary
new ideas ranged from outraged condemnation to fulsome praise.
Henri Poincaré called Cantor’s work a “grave disease”
that threatened to infect mathematics, and Leopold Kronecker
described Cantor as a “scientific charlatan” and a “corrupter
of youth.” Ludwig Wittgenstein, writing long after Cantor's death,
complained that mathematics had become “ridden through and through
with the pernicious idioms of set theory,” a theory he dismissed as
“utter nonsense,” “laughable,” and “wrong.”

At the other end
of the spectrum, in 1904, in the UK the Royal
Society awarded Cantor its highest award, the Sylvester
Medal, and in Germany David
Hilbert declared that “No one shall expel us from the Paradise
that Cantor has created.”

Having devoted
the early part of my professional career to work in (infinitary) set
theory, starting with my Ph.D. in “large cardinal theory,”
completed in 1971, and moving on to work on alternative universes of
sets (a particularly hot topic after Paul Cohen’s introduction of
the method of forcingin 1963), in
the early 1980s my interests started to shift elsewhere, to questions
about information, communication, and human reasoning.

Both discussions
raised the question as to whether study of infinity – in particular
the hierarchy of larger infinities that Cantor bequeathed to us –
would ever have any practical applications. As panelist Hugh Woodin
remarked at one point in the discussion, it is a foolish
mathematician who declares that a particular piece of mathematics
will not find applications. For instance, G. H. Hardy’s famous
statement (in his book A
Mathematician’s Apology) that his
work in number theory would never find practical application, proved
to be spectacularly wrong less than a century later, when number
theory became the foundation for internet security.

Hardy’s
observation was based on his familiarity of the world he lived in, a
world in which the World Wide Web was not even a dream. Today, we
cannot know what the world of tomorrow will look like. On the other
hand, whatever our children and grandchildren will take for granted,
their world will surely be finite, which makes it unlikely that
Cantor’s theory – and the almost a century of development in set
theory since then – will have practical use.

Or does it? What
about calculus? Infinitesimal (!) calculus not only has applications
in the modern world, but much of the science, technology, medicine,
and even financial structure the underpins our world depends on
calculus for its very existence. Applications don’t get more real
than that.

True, but the
dependence on infinities you find in calculus is essentially
asymptotic. What really drives calculus is the unending nature of
certain processes on the natural numbers. Talk of “infinitely
large” or “infinitely small” is little more than a manner of
speaking. Indeed, the epsilon-delta definitions (which do not involve
infinities or infinitesimals) are a way to formalize that manner of
speaking, effectively eliminating any actual infinite or
infinitesimal quantities.

In contrast, much
of the work on infinity (more precisely, infinities) carried out in
the second half of the twentieth century (when I was working in that
area) focused on properties of sets that made their cardinalities
super-infinities of different orders: inaccessible cardinals, Ramsey
cardinals, measurable cardinals, compact cardinals, supercompact
cardinals, Woodin cardinals, and so on. Infinities which dwarfed into
invisibility the puny cardinality of the set of natural numbers.
Indeed, each one in that sequence dwarfed all its predecessors into
invisibility. How could that work find an application?

I’ll lay my
cards on the table. I think the chances are that it won’t. But I don’t
think it is impossible. Indeed, I began to suspect a possible
application in the very domain I worked in after I left set theory.

[This may of
course be nothing more than a reflection of having at my disposal a
large hammer which made everything look a bit like a nail. But let’s
press on.]

The post 9/11
world saw me involved in a series of Defense Department projects the
first being improving intelligence analysis (and the others
essentially variants of that).

In today’s
information rich world, the major nations can be assumed to have
access to all the information they need to predict (and hopefully
thence prevent) the majority of terrorist attacks. The trouble is,
the few data points which must be identified and connected together
to determine the likelihood of a future attack are just a tiny few in
an overwhelming ocean of data. Even in the era of cloud computing,
identifying the key information is analogous to using the naked eye
to find a handful of proverbial needles in a non-proverbial field of
haystacks.

To all intents
and purposes, the available data is infinite. The only hope is to
impose some structure on the data that makes it possible for humans
and computers to work together on it, narrowing down the focus to the
regions more likely to be of significance. Though modern computing
systems can sift through massive (finite) amounts of data in a
relatively short time, they need to be programmed, and writing those
programs (at least, some kinds of them) will require some structure
on those large sets of data. A possible place to find the appropriate
structure(s) is infinitary set theory. In other words, to develop the
appropriate structures, assume the data is infinite. View the
infinite as a theoretical simplification of the very large finite.
(Economists sometimes make a similar simplifying assumption about
economies.)

Do I think this
is likely? Frankly, no. But then, neither could Hardy conceive of any
practical application of his work in number theory. [Incidentally,
like Hardy, I don’t think mathematics needs applications to justify
itself. It’s just that the question of application is what this
article is about!]

The discussion
about large cardinals you will find in those panel discussions at the
World Science Festival might seem impossibly abstract and far removed
from the everyday world. Indeed, it is. But the questions being
discussed all resulted from a process of rigorous, logical
investigation that arose directly from late nineteenth century
attempts to understand heat flow. History tells us that what begins
in the real world, very often ends up being used in the real world.

Prediction is
hard, particularly about the future.

Incidentally, how
did I end up working on a project for the DoD? They asked me. I might
not be the only person to speculate about a possible use of Cantor’s
paradise. This is your taxpayer dollars at work.

Wednesday, May 1, 2013

This month’s column is short, but I
am asking you to set aside 51 minutes and 36 seconds to watch the
embedded video. It is a recording of the Iris
M. Carl Equity Address given on Friday April 19 at this year’s NCTM
Annual Conference in Denver, Colorado. The title of the talk is
“Keeping Our Eyes on the Prize” and the
speaker is Uri Treisman, professor of
mathematics and of public affairs, and director of the Charles A.
Dana Center, at the University of Texas at Austin.

I was not able to
be at NCTM, but on the recommendation of several colleagues, I
watched the YouTube video. I simply cannot write a column on
mathematics or mathematics education in the same month as Treisman’s
immensely more important, profound—and powerfully articulated—words became part of mathematics education history. As a community,
we now have our own “I have a dream” speech. Thank you, Uri.

Monday, April 1, 2013

ADDED MAY 1: NOTE THAT THIS COLUMN WAS POSTED ON APRIL 1, "ALL FOOLS DAY" IN THE USA AND SEVERAL OTHER COUNTRIES.

One of the benefits of being at a university like Stanford is that we occasionally get the opportunity to see up close the emergence of an
amazing mathematical talent—someone who may turn out to be the next Euler or
Gauss.

Just over 18 months ago, Avril Wan was, to all
appearances, just another bright fourteen-year-old living in Taiwan, where her
father Yewful Wan runs a large shipping company and her Welsh-born mother
Melanie Wan is a university mathematics professor (and a former student of Timothy Gowers in Cambridge).

Then, in September 2011, Stanford computer science professor
Sebastian Thrun and Google researcher Peter Norvig offered what turned out to
be the first of what is now a flood of Massively Open Online Courses (MOOCs),
which make advanced university courses available to the entire world over the
Internet. Ms. Wan enrolled for that first MOOC, in artificial intelligence, and was the only student
to score a perfect 100% for the course.

When initial investigations made it clear that Ms. Wan’s
performance was legitimate, Thrun moved quickly, and arranged for Stanford to
offer her a place in Stanford’s famed Symbolic
Systems Program (which has produced a whole string of graduates who have
founded and led successful Silicon Valley companies, such as Reid Hoffman, who
founded LinkedIn, and Marissa Meyer, an early employee of Google and the new—and controversial—CEO of Yahoo!).

By the time Wan arrived at Stanford, Thrun had left to form Udacity, a Silicon Valley start-up offering
free online courses to the world, and the newly arrived student, who had just
turned 15 (and was accompanied by her mother), was assigned to the educational
care of another famous Stanford mathematics professor, Persi Diaconis, known
for his ability to see familiar problems in novel ways.

In late spring of 2012, there was a buzz across the Palo
Alto campus when it seemed that, under minimal guidance from Diaconis, the
young Ms. Wan had solved the notorious P = NP problem, but
Ron Graham of the University
of California at San Diego quickly found an error, pointing out that she had
implicitly assumed the existence of a complete, two-valued measure on the power
set of the natural numbers—a question first raised by the famous (Second
World) Wartime mathematician Stan Ulam.

Meanwhile, Ms. Wan’s mathematics blog had started to attract
attention back in her home country, making her somewhat of a Taiwan celebrity.
In particular, motivational videos she had posted on YouTube to encourage young
Taiwanese girls to study mathematics, eventually came to the attention of News
Corporation’s Rupert Murdoch, who pledged $5M to make her videos available
throughout the developing world.

But then, online tech journalist Dan Gillmor posted an article pointing out
that Murdoch’s funding was contingent on the distribution being restricted to
streaming to tablets supplied by his own, for-profit company Amplify. If so, that would surely have
killed the deal, since Ms. Wan recognizes the value of free educational
resources to the development of the less affluent countries of the world.

At that point, events started to unfold at the kind of
breakneck speed that only happens in Silicon Valley. Ms. Wan, still just 15 years old, remember, and technically without even a high school diploma, found herself inside the Palo Alto offices of the famed venture capital
company Greylock Partners, which was
willing to commit $100M to fund the establishment of a global, free, online
mathematics education platform, tentatively called “Wan World.”

With Greylock having been early stage funders of some of the
most successful start-up companies in recent years, most of which required
several years before anyone had the faintest idea how they would make money,
that interest was all it took to unleash the floodgates. Within a few days, Ms. Wan (or rather, the group of advisers her father quickly assembled to cope with
the interest) had been approached by Apple, Google, and Facebook, each of which
wanted to develop the platform on which Wan World would run, and by McGraw
Hill, Pearson Education, and Amazon, who wanted to own the content.

Meanwhile, despite all this frenzy, Ms. Wan herself seems
remarkably unfazed by the sudden changes in her life. Speaking to an unusually
full room in a recent meeting of Stanford’s Education’s
Digital Future lecture/discussion series (which is where I first met her),
she concluded her presentation by admitting to her fellow students, “Like you,
right now, I just want to graduate.”

With the second edition of my Stanford MOOC Introduction to Mathematical
Thinking starting this weekend on Coursera, I have once again been
wrestling with the question of the degree to which good, effective mathematics
learning can be achieved at scale, over the Internet.

Once I had made the decision to try to take (elements of) my
35-year-old mathematics transition course into the then emerging MOOC
format—less than a year ago!—I was
immediately brought face-to-face with the necessity of making use of two
educational devices I had loathed (and never used) throughout my entire career
in higher education:

machine-graded pop quizzes

machine-graded multiple-choice questions

For MAA readers, I don’t think I need to explain my dislike
for either of these über-simplistic devices, which can surely be justified in a
regular classroom only in terms of making life easier for the instructor.

Simply putting a class online does not require the use of
either device, of course. Technologies such as video conferencing and screen
sharing can make learning at a distance almost as good as traditional classroom
learning, and in some circumstances can make it better in some respects. But making a class available to tens of thousands of students online
changes everything. With such large numbers, the “class” dynamics change
dramatically. But it’s not all for the worse.

The first thing to realize is that a MOOC is in many ways
like radio or TV. Though both of those familiar features of modern life are
referred to as “mass media,” they are in fact highly individual. The newsreader
on radio or TV is not addressing a large audience; she or he is talking to
millions of single individuals. The
secret to being good on the radio or TV is to forget the millions and think of
just one (generic) person. After all, the listener or viewer is not in a room
with millions of other people; in fact, if the broadcast is successful, that
listener or viewer is cognitively in a room with
just the presenter. The really successful radio and TV newsreaders and
presenters are the ones who can do that really well. They create that sense
that they are talking just to You.

In my own case, I already knew that from many years of
occasional media work, but I think all MOOC instructors come to that
realization very quickly. When your voice, with or without your face, is in
someone’s living room, there is a direct human connection that in important
ways is far more intimate than is possible in a lecture hall filled with
anything more than a handful of students.

Once you realize this feature of the MOOC medium, the
underlying pedagogic model is obvious. It’s one-on-one teaching/learning—something that in the traditional academy is (of necessity) reserved only for
doctoral students.

At which point, the appropriate use of both pop quizzes and
multiple-choice questions starts to look feasible. (They ought to; doctoral
advisers use both extensively, and to great positive effect, though they do not
refer to them as such, and there is no machine-grading!)

Of course, in a MOOC it remains the case that the student
cannot communicate directly with the professor, nor can the professor see and
comment on an individual student’s work. That means two further techniques have
to be used as well:

peer tutoring

peer evaluation

In the first version of my MOOC, last September, I built the
course around the doctoral-student education model, deliberately setting out to
create the experience of a student sitting alongside me at my desk. (There is a
low resolution example here.)

But as a result of a career-long dislike of the first two
and a deep suspicion of the fourth, I used all but the third of those auxiliary
devices reluctantly and as little as possible. (The one I did embrace, peer
tutoring, did not work well the way I set it up. See below for details of Attempt
Two.)

Because of my caution, I think I avoided a fate reminiscent
of NASA’s first attempts to launch a rocket into space. But that was a first,
exploratory experience, and I wanted to live to try again. This time around,
based on what I learned, I am going to use all four much more aggressively, but
in ways I think might work.

I’ll be describing how I’ll be using them in a series of
posts to my blog MOOCtalk.org. For a brief—and decidedly limited—foretaste, check out this video excerpt of a
conversation my MOOC TA Paul Franz and I had recently with radio and TV
personality Angie Coiro, host of the syndicated radio and television
interview show In Deep.

The goal of Version 2 of the course is not to
reach the Moon. Chances are high that we’ll crash and burn. The goal is to at least
get off the ground before we do, and, if we are lucky, maybe even reach the
upper atmosphere. For sure, there will still be a long way to go.

If you want to live dangerously and be part of
this huge experiment, and if you have a Ph.D. (or pending Ph.D.) in mathematics and
several years of college teaching behind you, I am still looking for well
qualified volunteers to act as “Community TAs” for the course, to answer
students' questions on the course discussion forums. So far I have 14
volunteers, comprising 5 college professors, 3 Ph.D. students, 3
individuals currently working in the software industry, a K-12 education
consultant, a research laboratory scientist, and a stock analyst. If you want
to volunteer, and have the requisite experience, please drop me an email at devlin@stanford.edu. (There is no payment
for doing this—that includes me!) But being part of a large and truly global
community, who come together for several weeks for the sole purpose of
learning how to think mathematically (the course carries no college credit), is
truly a wonderful experience.

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.